A computationally universal phase of quantum matter Robert - - PowerPoint PPT Presentation

Robert Raussendorf, UBC

A computationally universal phase

• f

quantum matter

joint work with D.-S. Wang, D.T. Stephen, C. Okay, and H.P. Nautrup

The liquid phase of water

A quantum phase of spins

... which supports universal quantum computation

parameter 1 parameter 2

phase boundary phase

We consider:

• Phases of unique ground states of spin Hamiltonians, at T = 0
• In the presence of symmetry

Computational phases of quantum matter

parameter 1 2 r e t e m a r a p

phase boundary phase

physical characterization mathematical characterization group cohomology computational characterization MBQC power

A quantum phase of spins in 2D

... which supports universal quantum computation

parameter 1 parameter 2

phase boundary phase

We show:

• There exists a quantum phase of matter which is uni-

versal for quantum computation

• The computational power is uniform across the phase.
• Employ measurement-based quantum computation

Outline

• 1. “Computational phases of quantum matter”:
• Our motivation
• A short history of the question (1D)
• 2. A computationally universal phase of matter in 2D

Part I:

A short history of “computational phases of quantum matter”

Measurement-based quantum computation

Z X Y Z X Y Unitary transformation Projective measurement deterministic, reversible probabilistic, irreversible p 1-p

Measurement-based quantum computation

measurement of Z (⊙), X (↑), cos α X + sin α Y (ր)

• Information written onto the resource state, pro-

cessed and read out by one-qubit measurements only.

• Universal computational resources exist:

cluster state, AKLT state.

• R. Raussendorf, H.-J. Briegel, Physical Review Letters 86, 5188 (2001).

Motivation #1: MBQC and symmetry

ln(α)

1

N / N 1/N 1/ N

Can MBQC schemes be classified by symmetry, in a similar way as, say, elementary particles can?

An observation in quantum error-correction

quantum register

environment

Entanglement (good)

Entanglement (bad)

There’s good and bad entanglement. Good entanglement often comes with a symmetry

Motivation #2

How rare are MBQC resource states?

• 1. MBQC resource states are rare

Fraction of useful states smaller than exp(-n ) [n: number of qubits]

2

• D. Gross, S.T. Flammia, J. Eisert, PRL 2009.

• 1. MBQC resource states are rare

Fraction of useful states smaller than exp(-n ) [n: number of qubits]

2

T

• e

n t a n g l e d t

• b

e u s e f u l

• D. Gross, S.T. Flammia, J. Eisert, PRL 2009.

What about systems with symmetry?

In the presence of symmetry

• Computational power is uniform across physical phases

(known in 1D, conjectured beyond).

• Computationally useful quantum states are no longer rare.

Symmetry-protected topological order

Definition of SPT phases:

parameter 1 p a r a m e t e r 2

We consider ground states of Hamiltonians that are invariant under a symmetry group G.

Symmetry-protected topological order

parameter 1 p a r a m e t e r 2

Two points in parameter space lie in the same SPT phase iff they can be connected by a path of Hamiltonians such that

• 1. At every point on the path, the corresponding Hamiltonian is

invariant under G.

• 2. Along the path the energy gap never closes.

• 2. Symmetry protects computation
• A. Miyake, Phys. Rev. Lett. 105, 040501 (2010).

• 3. Symmetry-protected wire in MBQC
• Computational wire persists throughout symmetry-protected

phases in 1D.

• Imports group cohomology from the classification of SPT

phases.

D.V. Else, I. Schwartz, S.D. Bartlett and A.C. Doherty, PRL 108 (2012).

• F. Pollmann et al., PRB B 81, 064439 (2010); N. Schuch, D. Perez-Garcia, and I. Cirac,

PRB 84, 165139 (2011); X. Chen, Z.-C. Gu, and X.-G. Wen, PRB 83, 035107 (2011).

• 4. The SPT⇒MBQC meat grinder

G, [ω]

Lie group of gates for MBQC

quantum phases quantum computation

Hints at the classification of MBQC schemes by symmetry.

• J. Miller and A. Miyake, Phys. Rev. Lett. 114, 120506 (2015) [first 1D comp. phase].
• A. Prakash and T.-C. Wei, Phys. Rev. A (2016) [Wigner Eckart Theorem for MBQC].

RR, A.Prakash, D.-S. Wang, T.-C.Wei, D.T. Stephen, Phys. Rev. A (2017) [meat grinder].

Symmetry’s work and asymmetry’s contribution

ln(α)

1

N /N 1/N 1/ N

In 1D (at least):

• MBQC schemes classified by symmetry
• MBQC schemes operated using symmetry breaking

Inspection

The above waypoints 2 - 4 are about 1D systems. 1D is not sufficient for universal MBQC here is why:

• MBQC in spatial dimension D maps to the circuit model

in dimension D − 1 ⇒ Require D ≥ 2 for universality.

Are there computationally universal quantum phases in two dimensions? This talk describes one.

Part II:

A computationally universal SPT phase in 2D

Description of the 2D phase & result

• The symmetries of the phase are

X X X X X X X X X X X X

• The 2D cluster state is inside the phase
• Result. For a spin-1/2 lattice on a torus with circumferences n

and Nn, with n even, all ground states in the 2D cluster phase, except a possible set of measure zero, are universal resources for measurement-based quantum computation on n/2 logical qubits.

Consider MBQC resource states as tensor networks

Cluster-like states

... have PEPS tensors with the following symmetries

= = = = Z Z I I Z X I I X Z X Z X I I I Z X I I I

The cluster states have the additional symmetry

Z X

=

I

(We do not require the latter symmetry for cluster-like states)

Splitting the problem into halves

Part A: Lemma 1. All states in the 2D cluster phase are cluster-like. Part B: Lemma 2. All cluster-like states, except a set of measure zero, are universal for MBQC.

Part A: PEPS tensor symmetries

The physical symmetries

X X X X X X X X X X X X

in the 2D cluster phase imply the local PEPS tensor symmetries,

X Z X Z X Z X Z X Z Z = = = =

A: In cluster phase ⇒ cluster-like

Lemma 3. [*] Symmetric gapped ground states in the same SPT phase are connected by symmetric local quantum circuits

• f constant depth.

For any state |Φ in the phase, |Φ = UkUk−1..U1 |2D cluster. Look at an individual symmetry-respecting gate in the circuit, U =

• j

cjTj, with Tj ∈ P. Which Pauli observables Tj can be admitted in the expansion?

[*] X. Chen, Z.C. Gu, and X.G. Wen, Phys. Rev. B 82, 155138 (2010).

A: In cluster phase ⇒ cluster-like

Which Paulis Tj can be admitted in the expansion U =

j cjTj?

Z Z α β

i j

X Z Z I Z Z

A: In cluster phase ⇒ cluster-like

Which Paulis Tj can be admitted in the expansion U =

j cjTj?

Z Z α β

i j

X Z Z I Z Z non-local equivalent

multiply by Z Z X Z Z

Only X-type Pauli operators survive in the expansion.

Description of the local tensors:

C

=

A

Φ

B

Φ

a b c d

equivalent

• perators

equivalent

• perators

With

=

BΦ X BΦ X

Hence

=

C BΦ

=

C BΦ AΦ

=

AΦ

X X I Z I X I Z I X X X Z Z =

C BΦ

X X X Z Z

• Local tensors AΦ describing |Φ are invariant under the cluster-

like symmetries.

Between Parts A and B

virtual quantum register virtual circuit time

• The “virtual” quantum register is located on the horizontal

tensor legs

• D. Gross and J. Eisert, Phys. Rev. Lett. 98, 220503 (2007).

Part B: Symmetry Lego

Just shown: PEPS tensor symmetries hold throughout the 2D cluster phase

X Z X Z X Z X Z X Z Z = = = =

• Now weave them into larger patterns.

B: Cluster-like ⇒ universal

The clock cycle:

Z Z

v i r t u a l q u a n t u m r e g i s t e r circuit model time X X X X X X X X X

X

virtual quantum register circuit model time

X X

X X X X X X X X

• Every logical operator is mapped back to itself after n

columns (n = circumference). ⇒ This defines the clock cycle for gate operation.

B: Cluster-like ⇒ universal

2D-locality (measurement) quasi-1D locality (clock cycle) virtual quantum register

• Map 2D system to effective 1D system

B: Cluster-like ⇒ universal

eidα Zk eidα Xk−1ZkXk+1 eidα Xk Universal gate set on n/2 qubits

B: Cluster-like ⇒ universal

2D cluster state: eidα Zk eidα Xk−1ZkXk+1 eidα Xk Throughout the phase: ei|ν|dα Zk ei|ν|dα Xk−1ZkXk+1 ei|ν|dα Xk |ν| ≤ 1 (ν depends on the location in the phase)

About ν: RR, A.Prakash, D.-S. Wang, T.-C.Wei, D.T. Stephen, Phys. Rev. A (2017).

Result

• The symmetries of the phase are

X X X X X X X X X X X X

• The 2D cluster state is inside the phase
• Result. For a spin-1/2 lattice on a torus with circumferences n

and Nn, with n even, all ground states in the 2D cluster phase, except a possible set of measure zero, are universal resources for measurement-based quantum computation on n/2 logical qubits.

Summary and outlook

• There exists a symmetry-protected phase in 2D with uniform

universal computational power for MBQC.

• Goal: Classification of MBQC schemes by symmetry.
• Symmetry Lego is fun—Try it!
• Phys. Rev. Lett. 122, 090501 (2019)

Also see: Quantum 3, 142 (2019)

X Z X Z X Z X Z X Z Z = = = =

The parameter ν

There is a complex-valued parameter ν, |ν| ≤ 1, that needs to be known about the location of the resource state within the phase.

Deviate from protected basis dβ

For infinitesimal angles dβ, this results in a logical rotation [*] eidβ|ν| T, for some Pauli operator T. (E.g., T = Zk, Xk, Xk−1ZkXk+1). We require that ν = 0.

[*] RR, D.-S. Wang, A. Prakash, T.-C. Wei, D.T. Stephen, PRA 96 (2017).